A Categorical Development of Right Derived Functors
Skyler Marks
TL;DR
This paper presents a purely categorical construction of homological algebra, focusing on right derived functors within abelian categories. It builds from additive categories defined via biproducts to cochain complexes, defining cohomology objects $H^n(A^\bullet)$ and establishing the abelian structure of ${\mathbf{CCH}}(\mathscr A)$. Injective resolutions are introduced under the assumption of enough injectives, enabling the definition of right derived functors $R^iF$ as $R^iF(A)=H^i(F(I^\bullet))$ and ensuring well-definedness, functoriality, and resolution-independence through comparison lemmas. The framework captures central constructions like Ext in a fully categorical, minimal setting, highlighting the feasibility of deriving homological invariants without relying on concrete algebraic categories. Overall, the work provides a compact, rigorous foundation for deriving functors in category theory and demonstrates the coherence of cohomological methods within abelian categories.
Abstract
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm of pure abstract category theory in their development of the field, leveraging the Freyd-Mitchell embedding theorem or similar results, or otherwise using set-theoretic language to augment a general categorical discussion. This paper seeks to demonstrate that - while it is not necessary for most mathematicians' purposes - a development of homological concepts can be contrived from purely categorical notions. We begin by outlining the categories we will work within, namely Abelian categories (building off additive categories). We continue to develop cohomology groups of sequences, eventually culminating in a development of right derived functors. This paper is designed to be a minimalist construction, supplying no examples or motivation beyond what is necessary to develop the ideas presented.